Math 650-600, Several Complex Variables, Fall 2007
Harold P. Boas

General course information (the first-day handout is available also in pdf format)
Lecture notes (7 December 2007)

What's new

Friday, December 7: I posted an update to the lecture notes.
Monday, December 3: We proved the basic estimate on bounded pseudoconvex domains with smooth boundary, thus completing the solution of the d-bar-problem.
Friday, November 30: We set up the basic L² estimate and used it to solve the d-bar-problem for (0,1)-forms on bounded pseudoconvex domains.
Wednesday, November 28: We solved the Levi problem for bounded, strongly pseudoconvex domains with smooth boundary, granted the solvability of the d-bar equation.
Monday, November 26: We proved that every pseudoconvex domain can be exhausted by an increasing sequence of smoothly bounded, strongly pseudoconvex domains (domains that are locally holomorphically equivalent to strongly convex domains).
Wednesday, November 21: We proved that Levi pseudoconvexity implies pseudoconvexity (in the sense of the existence of a plurisubharmonic exhaustion function).
Monday, November 19: We introduced the notion of Levi pseudoconvexity and proved that it is a consequence of pseudoconvexity (in the sense that the negative of the logarithm of the distance to the boundary is plurisubharmonic).
Friday, November 16: We proved that pseudoconvexity is a local property of the boundary of a domain.
Wednesday, November 14: We completed the proof of the equivalence of four notions of pseudoconvexity.
Monday, November 12: We proved that the Kontinuitätssatz implies that the negative of the logarithm of the distance to the boundary is plurisubharmonic, and we set up the proof that the latter property implies the existence of an infinitely differentiable, strictly plurisubharmonic exhaustion function.
Sunday, November 11: I posted an updated version of the lecture notes.
Friday, November 9: We proved some of the equivalences among properties characterizing pseudoconvexity.
Wednesday, November 7: We followed up on some past exercises, corrected an erroneous statement, and posed the problem of characterizing sub-pluriharmonic functions (which are a superset of the plurisubharmonic functions).
Monday, November 5: We solved two of the outstanding exercises about domains of holomorphy.
Friday, November 2: We proved that various properties are equivalent to plurisubharmonicity.
Wednesday, October 31: We discussed properties equivalent to plurisubharmonicity and looked at some examples.
Monday, October 29: We discussed solutions to some of the outstanding exercises.
Sunday, October 28: I posted an update to the lecture notes.
Friday, October 26: We continued the discussion of subharmonic functions and previewed the statement of the Levi problem.
Wednesday, October 24: We discussed lattice properties of subharmonic functions and envelopes of families of subharmonic functions.
Monday, October 22: We discussed some properties of subharmonic functions in the plane.
Friday, October 19: I posed some exercises about domains of holomorphy, and I stated some facts about subharmonic functions in preparation for discussing the notion of pseudoconvexity.
Thursday, October 18: I posted an update to the lecture notes.
Wednesday, October 17: We finished the proof that a domain is holomorphically convex if and only if it is a domain of holomorphy.
Monday, October 15: We proved that a weak domain of holomorphy is holomorphically convex, and we started the proof that most holomorphic functions on a holomorphically convex domain are singular at every boundary point.
Friday, October 12: We continued discussing the notion of domain of holomorphy, stated a theorem showing among other things that a weak domain of holomorphy is a domain of holomorphy, and began the proof.
Wednesday, October 10: We finished the proof from last time except for the equivalence with the notion of a domain of holomorphy, which we (finally) defined precisely.
Monday, October 8: We started proving the Cartan-Thullen theorem characterizing holomorphically convex domains.
Friday, October 5: Since I was speaking in Chicago, Michael Fulkerson discussed automorphisms of the ball and related topics.
Wednesday, October 3: Students presented solutions to two of the exercises, and we saw why the closure of the Hartogs triangle is not rationally convex. On Friday, I will be speaking at an AMS meeting in Chicago, and Michael Fulkerson will talk in class about automorphisms of the ball and the polydisc.
Monday, October 1: We showed that rational convexity can be rephrased as a weak polynomial separation property, and we looked at the example of the Hartogs triangle.
Sunday, September 30: I posted an update to the lecture notes.
Friday, September 28: We characterized domains convex with respect to linear fractional functions by the existence of supporting complex hyperplanes at boundary points, and we introduced the notion of rational convexity.
Wednesday, September 26: We repaired the proof from last time and began discussing convexity with respect to linear fractional functions.
Monday, September 24: We attempted to prove that every polynomially convex set can be arbitrarily well approximated by polynomial polyhedra.
Friday, September 21: We discussed some additional examples of polynomially convex sets, such as polynomial polyhedra and graphs of polynomials.
Wednesday, September 19: We worked out some examples of polynomially convex sets. In particular, we observed that polynomial convexity is characterized by a topological property in dimension one but not in higher dimensions.
Monday, September 17: We solved the exercise about convexity with respect to absolute values of complex linear or affine complex linear functions, and we started discussing polynomial convexity.
Friday, September 14: We continued the discussion of convexity conditions and showed that convexity with respect to the monomials is equivalent to logarithmic convexity.
Wednesday, September 12: We discussed the notion of convexity with respect to a class of functions.
Monday, September 10: We completed the proof from last time that every logarithmically convex complete Reinhardt domain is the domain of convergence of some power series.
Friday, September 7: We discussed Exercise 4 about Fatou-Bieberbach domains and then proved (modulo some details) that logarithmically convex complete Reinhardt domains are convergence domains for power series.
Wednesday, September 5: Students presented solutions to the first three exercises, and we proved Pringsheim's lemma, thereby completing the proof from last time that a convergence domain is a domain of holomorphy.
Monday, September 3: We used the Baire category theorem to prove that a convergence domain for a power series is a domain of holomorphy.
Friday, August 31: We discussed some properties of convergence domains of multi-variable power series.
Wednesday, August 29: We continued the discussion of differences between one and several complex variables. The inhomogeneous Cauchy-Riemann equations form an overdetermined system in dimensions greater than 1, and when the compatibility condition for solvability is satisfied, one can solve with compact support if the data have compact support; the situation in dimension 1 is different. In contrast to the one-dimensional Riemann mapping theorem, there is no topological property in higher dimensions that detects when a domain is biholomorphically equivalent to a ball.
Monday, August 27: There was confusion about the room, which the registrar changed from MILS 213 to HALB 104. After most of us got to the right place, we discussed some ways in which multi-dimensional complex analysis differs from one-dimensional function theory. For example, there are infinitely many different types of convergence domains for multi-variable power series, and there is no multi-dimensional analogue of the Cauchy integral having a kernel that is both universal (independent of the domain) and holomorphic in the free variable.
Friday, August 24: This site went live today. The first-day handout is available online.

Math 650-600, Several Complex Variables, Fall 2007 Harold P. Boas

What's new

Math 650-600, Several Complex Variables, Fall 2007
Harold P. Boas